1.  mgmg FNFN x FNFN mgmg   fkfk fkfk s s  If no net non-conservative forces  Then, conservation of mechanical energy holds Crate on Incline Revisited FBD h

2.  The crate starts from rest, v 0 =0  Some energy, W NC is loss from the system  In this case it is due to the non-conservative friction force  energy loss in the form of heat

3.  Because of friction, the final speed is only 9.3 m/s as we found earlier  If the incline is frictionless, the final speed would be:  Because of the loss of energy, due to friction, the final velocity is reduced. It seems that energy is not conserved

4. Conservation of Energy  There is an overall principle of conservation of energy  Unlike the principle of conservation of mechanical energy, which can be broken, this principle can not  It says: ``The total energy of the universe is, has always been, and always will be constant. Energy can neither be created nor destroyed, only converted from one form to another.’’  So far, we have only been concerned with mechanical energy

5.  There are other forms of energy: heat, electromagnetic, chemical, nuclear, rest mass (E m =mc 2 )  Q (W NC ) is the energy lost (or gained) by the mechanical system

6.  The electrical utility industry does not produce energy, but merely converts energy Lake River Hydro-power plant h PE=mgh KE=mv 2 /2 electricity Light, heat Power Average power: Units of J/s=Watt (W) Measures the rate at which work is being done

7. Example A car accelerates uniformly from rest to 27 m/s in 7.0 s along a level stretch of road. Ignoring friction, determine the average power required to accelerate the car if (a) the weight of the car is 1.2x10 4 N, and (b) the weight of the car is 1.6x10 4 N. Solution: Given: v 0 =0, v f =27 m/s,  t=7.0 s, (a) mg= 1.2x10 4 N, (b) mg= 1.6x10 4 N Method: determine the acceleration

8.  We don’t know the displacement s  The car’s motor provides the force F to accelerate the car – F and s point in same direction Need a s s

9. (a) (b) Or from work-energy theorem Same as on previous slide

10. Example: Problem 6.77 A particle starting from point A, is projected down the curved runway. Upon leaving the runway at point B, the particle is traveling straight upward and reaches a height of 4.00 m above the floor before falling back down. Ignoring friction and air resistance, find the speed of the particle at point A. 3.00 m 4.00 m A